Question

a. A programmer is taking a two-hours-time-limit makeup examination. Suppose the probability
that the programmer will finish the exam at most y hours is y/3, for all 0 ≤ y ≤ 2. Given that the
student is still working after 1.75 hours, what is the conditional probability that the full time is
used?
b. A multiple choice test has 7 questions with 3 wrong choices and 1 correct choice each. How many
ways are there to answer the test? What is the probability that two papers have the same answers?

Answer 1

Suppose the student finishes the exam at time $t.$

$$P(t<y)=y/3, \ 0\leq y\leq2$$

The 1conditional probability that the full time is used will be

$$P(t>2|t>1.75)=\dfrac{P(t>2\cap t>1.75)}{P(t>1.75)}=$$

$$=\dfrac{P(t>2)}{P(t>1.75)}=\dfrac{1-P(t\leq2)}{1-P(t\leq1.75)}=$$

$$=\dfrac{1-2/3}{1-1.75/3}=0.8$$

b. There are 4 choices per question, and 7 questions. So there are

$$4\times4\times4\times4\times4\times4\times4=4^7=16384$$

ways to answer.

What is the probability that two papers have the same answer to the first question?

$$p={1\over4}$$

What is the probability that two papers have the same answers ?

$$P(same)=({1\over 4})^7={1\over 16384}\approx0.000061$$